Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $n \neq 0$. $p = \dfrac{-5n - 45}{n^2 + n - 72} \div \dfrac{-8n - 40}{-3n + 24} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{-5n - 45}{n^2 + n - 72} \times \dfrac{-3n + 24}{-8n - 40} $ First factor the quadratic. $p = \dfrac{-5n - 45}{(n - 8)(n + 9)} \times \dfrac{-3n + 24}{-8n - 40} $ Then factor out any other terms. $p = \dfrac{-5(n + 9)}{(n - 8)(n + 9)} \times \dfrac{-3(n - 8)}{-8(n + 5)} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac{ -5(n + 9) \times -3(n - 8) } { (n - 8)(n + 9) \times -8(n + 5) } $ $p = \dfrac{ 15(n + 9)(n - 8)}{ -8(n - 8)(n + 9)(n + 5)} $ Notice that $(n + 9)$ and $(n - 8)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac{ 15(n + 9)\cancel{(n - 8)}}{ -8\cancel{(n - 8)}(n + 9)(n + 5)} $ We are dividing by $n - 8$ , so $n - 8 \neq 0$ Therefore, $n \neq 8$ $p = \dfrac{ 15\cancel{(n + 9)}\cancel{(n - 8)}}{ -8\cancel{(n - 8)}\cancel{(n + 9)}(n + 5)} $ We are dividing by $n + 9$ , so $n + 9 \neq 0$ Therefore, $n \neq -9$ $p = \dfrac{15}{-8(n + 5)} $ $p = \dfrac{-15}{8(n + 5)} ; \space n \neq 8 ; \space n \neq -9 $